# Hilbert–Burch theorem

In mathematics, the **Hilbert–Burch theorem** describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968, p.944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof.

## Statement

If *R* is a local ring with an ideal *I* and

is a free resolution of the *R*-module *R*/*I*, then *m* = *n* – 1 and the ideal *I* is *aJ* where *a* is a non zero divisor of *R* and *J* is the depth 2 ideal generated by the determinants of the minors of size *m* of the matrix of the map from *R*^{m} to *R*^{n}.

## References

- Burch, Lindsay (1968), "On ideals of finite homological dimension in local rings",
*Proc. Cambridge Philos. Soc.*,**64**: 941–948, doi:10.1017/S0305004100043620, ISSN 0008-1981, MR 0229634, Zbl 0172.32302 - Eisenbud, David (1995),
*Commutative algebra. With a view toward algebraic geometry*, Graduate Texts in Mathematics,**150**, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001 - Eisenbud, David (2005),
*The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry*, Graduate Texts in Mathematics,**229**, New York, NY: Springer-Verlag, ISBN 0-387-22215-4, Zbl 1066.14001 - Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen",
*Mathematische Annalen*(in German),**36**(4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831, JFM 22.0133.01

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